#### Abstract

The idea of -almost convergence (briefly, -convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm on such that it is a Banach space and then we define and characterize those four-dimensional matrices which transform -convergence of double sequences into -convergence. We also define a -core of and determine a Tauberian condition for core inclusions and core equivalence.

#### 1. Background, Notations, and Preliminaries

We begin by recalling the definition of convergence for double sequences which was introduced by Pringsheim [1]. A double sequence is said to be to in the Pringsheim's sense (or -convergent to ) if for given there exists an integer such that whenever . We will write this as where and are tending to infinity independent of each other. We denote by the space of -convergent sequences.

We say that a double sequence is if Denote by the space of all bounded double sequences.

If a double sequence in and is also -convergent to , then we say that it is* boundedly **-convergent* to (or, BP-convergent to ). We denote by the space of all boundedly -convergent double sequences. Note that .

We remark that, in contrast to the case for single sequences, a -convergent double sequence need not be bounded.

Let be a four-dimensional infinite matrix of real numbers for all and a space of double sequences. Let be a double sequences space, converging with respect to a convergence rule . Define Then, we say that a four-dimensional matrix maps the space into the space if and is denoted by .

The idea of almost convergence of Lorentz [2] is narrowly connected with the limits of Banach (see [3]) as follows. A sequence in is almost convergent to if all of its Banach limits are equal, where denotes the space of all bounded sequences. Mohiuddine [4] applies this concept to established some approximation theorems for sequence of positive linear operator. Móricz and Rhoades [5] extended the notion of almost convergence from single to double sequences as follows.

A double sequence of real numbers is said to be* almost convergent* to a number if
For more details on almost convergence for single and double sequences, one can refer to [6–13].

The two-dimensional analogue of Banach limit has been defined by Mursaleen and Mohiuddine [14] as follows. A linear functional on is said to be* Banach limit* if it has the following properties: if (i.e., for all ),, where with for all ,,where the shift operators , , and are defined by
Denote by the set of all Banach limits on . Note that if (B) holds, then we may also write . A double sequence is said to be* almost convergent* to a number if for all .

Let and be two nondecreasing sequences of positive reals and each tending to such that , , , , and
is called the* double generalized de la Valée-Poussin mean*, where and . We denote the set of all and type sequences by using the symbol .

Quite recently, Mohiuddine and Alotaibi [15] presented a generalization of the notion of almost convergent double sequence with the help of de la Vallée-Poussin mean and called it -almost convergent. In the same paper, they also defined and characterized some four-dimensional matrices. For more details on double sequences, four-dimensional matrices, and other related concepts, one can refer to [16–26].

A double sequence of reals is said to be -*almost convergent* (briefly, -convergent) [15] to some number if , where
Denote by the space of all almost convergent sequences . Note that .

We remark that if we take and , then the notion of -almost convergence coincides with the notion of almost convergence for double sequences due to Móricz and Rhoades [5].

#### 2. -Matrices

We will assume throughout this paper that the limit of a double sequence means limit in the Pringsheim sense. We define the following matrix classes and establish interesting results.

*Definition 1. *A four-dimensional matrix is said to be -*almost regular* if for all with , and one denotes this by .

*Definition 2. *A matrix is said to be of class if it maps every -convergent double sequence into -convergent double sequence; that is, for all . In addition, if , then is and, in symbol, one will write .

Now we define the norm on as follows.

Theorem 3. * is a Banach space normed by
*

*Proof. *It can be easily verified that (8) defines a norm on . We show that is complete. Now, let be a Cauchy sequence in . Then for each , is a Cauchy sequence in . Therefore (say). Put ; given there exists an integer , say, such that, for each ,
Hence
then, for each , , , and , , we have
Taking limit , we have for and for each of
Now for fixed , the above inequality holds. Since for fixed , , we get
uniformly in , . For given , there exist positive integers , such that
for , and for all , . Here , are independent of , but depend upon . Now by using (12) and (14), we get
for , and for all , . Hence and is complete.

Now we characterize the matrix class as well as . Let be the subspace of such that , uniformly in ; that is Note that every can be written as where uniformly in , and with for all .

Theorem 4. *A matrix if and only if *(A)*,*(A)*,*(A)*,**where is the shift operator.*

*Proof. **Necessity.* Let . We know that , so we have . Hence the necessity of (A) follows. Since , then . This is equivalent to
that is, (A) holds. For each , we have because
for all Banach limit . Hence ; that is, (A) holds.*Sufficiency.* Let conditions (A)–(A) hold and . Then
where , , uniformly in and with for all . Taking -transform in (20), we obtain
If , then by (A) we have . Since by (A), is bounded linear operator on , we get . This yields . Now from condition (A) and (21), . Therefore .

Corollary 5. *A matrix if and only if conditions (A) and (A) with and (A) hold.*

#### 3. Some Core Theorems

The core or Knopp core of a real number single sequence is the closed interval (see [27, 28]). In 1999, Patterson [29] extended the Knopp core from single sequences to double sequences and called it Pringsheim core (shortly, -core) which is given by . In the recent past, the -core and -core for double sequences have been defined and studied by Mursaleen and Edely [30] and Mursaleen and Mohiuddine [31, 32], respectively, while the -core for single sequences is given by Mishra et al. [33]. In 2011, Kayaduman and Çakan [34] presented the concept of Cesáro core of double sequences.

We define the following sublinear functional on : Then we define the -core of a real-valued bounded double sequence to be the closed interval .

Since every -convergent double sequence is -convergent, we have where , and hence it follows that for all .

Theorem 6. *For every ,
**
if and only if*(C)* is -regular,*(C)*,**where
*

*Proof. **Necessity.* Suppose that (24) holds for all . One obtains
that is,
If , then
that is,
Therefore is -regular. This yields the necessity of (C).

Now, with the help of Lemma 2.1 of [35], there is a double sequence such that and
If a double sequence defined by
then
where
This yields the necessity of (C). *Sufficiency.* We know that . Following the lines of Theorem 2 of [31] for translation mapping, one obtains
For any , we have
Taking infimum over , we obtain
Thus
Since , we can write
where , since is -regular. Operating to (38), one obtains
By -almost regularity, we have
From the definition of , we get
uniformly in . Also
Therefore we obtain from (40) that
uniformly in . Equations (37) and (43) give that
that is,
As , one obtains .

Note that . This motivates us to prove the following result by adding a condition to get a more general result.

Theorem 7. *For , if
**
holds, then .*

*Proof. *By the definition of -core and -core, we have to show that . Let . Then, for given , for all and for large it follows from the definition of that
We can write
Since (46) holds, for given , we get that
for all . Thus we have
Equation (49) yields
Taking in (48) and using (51), one obtains . Since is arbitrary, we obtain .

Corollary 8. *If (46) holds and is -convergent, then is convergent.*

Finally, we define the concepts of -almost uniformly positive and -almost absolutely equivalent and establish a theorem related to these concepts.

*Definition 9. *A matrix is said to be -*almost uniformly positive*, denoted by -uniformly positive, if

*Definition 10. *Let and be two -regular matrices and
Then and are said to be -*almost absolutely equivalent*, denoted by -absolutely equivalent, on whenever ; that is, either and both tend to the same -limit or neither of them tends to a -limit, but their difference tends to -limit zero.

Before proceeding further, first we state the following lemma which we will use to our next result.

Lemma 11. *For , if , then .*

Proof of the lemma is straightforward and thus omitted.

Theorem 12. *Let be a -regular matrix. Then, for all if and only if there is a -regular matrix such that is -uniformly positive and -absolutely equivalent with on .*

*Proof. *Let there be a -regular matrix such that is -uniformly positive and -absolutely equivalent with on . Then, by (53) and -absolutely equivalent of and , we have
uniformly in . Now, by Lemma 11, for all . By Theorem 6, we have , since is arbitrary.

Conversely, let for all . Then by Theorem 6, is -uniformly positive. Now we define a matrix as
for all , , , . Then it is easy to see that is -regular since is -regular, and
Further
Since is -regular, we have by (57) that
Thus is -uniformly positive. Further, it follows from (56) that and are -absolutely equivalent.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130-265-D1435). The authors, therefore, acknowledge with thanks DSR technical and financial support.